On 7/12/2016 10:55 AM, James Propp wrote:
Go to http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/ulamSpirals.h... , hover your mouse over "modulo 2 (low res) <http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/ulamSpirals.htm#nogo>", and step back. You should see a large "X" in the image. (With my glasses on, I see the "X" when I'm about five feet away from my laptop; with my glasss off, a foot will do.)
What's happening here?
That's really two questions. What is the image made of, on the pixel level? (I'm guessing that some of the "squares" are actually rectangles with aspect ratio close to 1.) And: What are my eyes and brain doing? (Why is the effect so salient from far away but invisible up close?)
Most of the "squares" are actual squares that are 5 pixels on a side. However, the four regions bounded by the diagonals are slightly out of phase with each other, each shifted one pixel left-right or up-down from its neighbors. On the boundaries this results in some "squares" which are only 4-by-5 (or 5-by-4) rectangles. Along three of the diagonals, these smaller rectangles are black, so those diagonals appear lighter; on the upper-left diagonal, the rectangles are white, so that diagonal is darker. And, right in the middle, there is a single 4-by-4 pixel black "square". A magnified view of the central portion of the image (9-by-9 "squares") with grid lines added can be seen at https://imgur.com/XqSzlpY . -- Fred W. Helenius fredh@ix.netcom.com